What do you advise me to do regarding my family's refusal to marry me to a committed girl because of her mother's old reputation, and how can I convince my stubborn mother with verses and hadiths?

The forbidden relationship with the girl must be severed immediately. Engagement does not permit a relationship between a man and a woman unless it is a marriage contract.

Regarding the mother's objection, one should be gentle with her and remind her of the importance of a lawful marriage, and that there is no harm in marrying a widow, and that this was common during the time of the Companions.

If the mother refuses absolutely and without a legitimate religious reason, it is advisable to forgo marrying this girl if the young man does not fear temptation for himself, for whoever gives up something for the sake of Allah, Allah will compensate him with something better. However, if he fears temptation for himself, and his mother does not have acceptable religious reasons, then it is permissible for him to marry her, while being keen to appease his mother after the marriage.